Universal Matrix Completion
This addresses the problem of matrix completion for practitioners by removing the need for uniform random sampling and fresh sampling per matrix, offering a more flexible and efficient solution.
The paper tackles the restrictive assumptions in low-rank matrix completion by providing a universal recovery guarantee that works for various sampling schemes, showing that nuclear norm minimization can exactly recover low-rank matrices under certain incoherence conditions with a sample complexity of O(nr^2) for uniformly sampled entries, compared to O(nr log n) in existing methods.
The problem of low-rank matrix completion has recently generated a lot of interest leading to several results that offer exact solutions to the problem. However, in order to do so, these methods make assumptions that can be quite restrictive in practice. More specifically, the methods assume that: a) the observed indices are sampled uniformly at random, and b) for every new matrix, the observed indices are sampled afresh. In this work, we address these issues by providing a universal recovery guarantee for matrix completion that works for a variety of sampling schemes. In particular, we show that if the set of sampled indices come from the edges of a bipartite graph with large spectral gap (i.e. gap between the first and the second singular value), then the nuclear norm minimization based method exactly recovers all low-rank matrices that satisfy certain incoherence properties. Moreover, we also show that under certain stricter incoherence conditions, $O(nr^2)$ uniformly sampled entries are enough to recover any rank-$r$ $n\times n$ matrix, in contrast to the $O(nr\log n)$ sample complexity required by other matrix completion algorithms as well as existing analyses of the nuclear norm method.